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### Open Access Journals

DCDS-B

A continuous map $f:[0,1]\rightarrow[0,1]$ is called an $n$-modal
map if there is a partition $0=z_0 < z_1 < ... < z_n=1$ such that
$f(z_{2i})=0$, $f(z_{2i+1})=1$ and, $f$ is (not necessarily strictly)
monotone on each $[z_{i},z_{i+1}]$. It is well-known that such a
map is topologically semi-conjugate to a piecewise linear map; however
here we prove that the topological semi-conjugacy is unique for this
class of maps; also our proof is constructive and yields a sequence
of easily computable piecewise linear maps which converges uniformly
to the semi-conjugacy. We also give equivalent conditions for the
semi-conjugacy to be a conjugacy as in Parry's theorem. Related work
was done by Fotiades and Boudourides and Banks, Dragan and Jones,
who however only considered cases where a conjugacy exists. Banks,
Dragan and Jones gave an algorithm to construct the conjugacy map
but only for one-hump maps.

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